Best Known (12, 24, s)-Nets in Base 81
(12, 24, 1094)-Net over F81 — Constructive and digital
Digital (12, 24, 1094)-net over F81, using
- net defined by OOA [i] based on linear OOA(8124, 1094, F81, 12, 12) (dual of [(1094, 12), 13104, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8124, 6564, F81, 12) (dual of [6564, 6540, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8124, 6564, F81, 12) (dual of [6564, 6540, 13]-code), using
(12, 24, 2394)-Net over F81 — Digital
Digital (12, 24, 2394)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8124, 2394, F81, 2, 12) (dual of [(2394, 2), 4764, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8124, 3283, F81, 2, 12) (dual of [(3283, 2), 6542, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(8124, 3283, F81, 2, 12) (dual of [(3283, 2), 6542, 13]-NRT-code), using
(12, 24, 1610910)-Net in Base 81 — Upper bound on s
There is no (12, 24, 1610911)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 6362 691596 349290 310830 510117 846645 015850 613281 > 8124 [i]